This review examines the biological background to the development of ideas on rapid eye movement sleep (REM sleep), so-called paradoxical sleep (PS), and its relation to dreaming. Aspects of the phenomenon which are discussed include physiological changes and their anatomical location, the effects of total and selective sleep deprivation in the human and animal, and REM sleep behavior disorder, the latter with its clinical manifestations in the human. Although dreaming also occurs in other sleep phases (non-REM or NREM sleep), in the human, there is a contingent relation between REM sleep and dreaming. Thus, REM is taken as a marker for dreaming and as REM is distributed ubiquitously throughout the mammalian class, it is suggested that other mammals also dream. It is suggested that the overall function of REM sleep/dreaming is more important than the content of the individual dream; its function is to place the dreamer protagonist/observer on the topographical world. This has importance for the developing infant who needs to develop a sense of self and separateness from the world which it requires to navigate and from which it is separated for long periods in sleep. Dreaming may also serve to maintain a sense of 'I'ness or "self" in the adult, in whom a fragility of this faculty is revealed in neurological disorders.